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CHAPTER 18

Routing Games

Tim Roughgarden

Abstract

This chapter studies the inefficiency of equilibria in noncooperative routing games, in which self-interested players route traffic through a congested network. Our goals are threefold: to introducethe most important models and examples of routing games; to survey optimal bounds on the price ofanarchy in these models; and to develop proof techniques that are useful for bounding the inefficiencyof equilibria in a range of applications.

18.1 Introduction

A majority of the current literature on the inefficiency of equilibria concerns routinggames. One reason for this popularity is that routing games shed light on an importantpractical problem: how to route traffic in a large communication network, such as theInternet, that has no central authority. The routing games studied in this chapter arerelevant for networks with “source routing,” in which each end user chooses a fullroute for its traffic, and also for networks in which traffic is routed in a distributed,congestion-sensitive manner. Section 18.6 contains further details on these applications.

This chapter focuses on two different models of routing games, although the in-efficiency of equilibria has been successfully quantified in a range of others (seeSection 18.6). The first model, nonatomic selfish routing, is a natural generalization ofPigou’s example (Example 17.1) to more complex networks. The modifier “nonatomic”refers to the assumption that there are a very large number of players, each controllinga negligible fraction of the overall traffic. We also study atomic selfish routing, whereeach player controls a nonnegligible amount of traffic. We single out these two modelsfor three reasons. First, both models are conceptually simple but quite general. Sec-ond, the price of anarchy is well understood in both of these models. Third, the twomodels are superficially similar, but different techniques are required to analyze theinefficiency of equilibria in each of them.

The chapter proceeds as follows. Section 18.2 introduces nonatomic and atomicselfish routing games and explores several examples. Section 18.3 studies the existence

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460 routing games

and uniqueness of equilibria in routing games. It also offers a glimpse of the potentialfunction method, a technique that will be developed further in Chapter 19. Section 18.4proves tight upper bounds on the price of anarchy in nonatomic and atomic selfishrouting games. Section 18.5 proposes two ways to reduce the price of anarchy innonatomic selfish routing games. Section 18.6 concludes with bibliographic notes.

18.2 Models and Examples

18.2.1 Nonatomic Selfish Routing

To introduce nonatomic selfish routing games, we recall the essential features ofPigou’s example (Example 17.1 and Figure 17.1). First, we are given a networkdescribing the routes available to the players. In Pigou’s example, there are two parallelroutes, each a single edge, that connect a source vertex s to a sink vertex t . Each edgehas a cost that is a function of the amount of traffic that uses the edge. We assume thatselfish players choose routes to minimize the cost incurred; in an equilibrium outcome,all players choose a path of minimum cost. In the equilibrium in Pigou’s example, allplayers choose the second edge, and the cost of this edge in this outcome is 1.

More generally, a selfish routing game occurs in a multicommodity flow network,or simply a network. A network is given by a directed graph G = (V, E), with vertexset V and directed edge set E, together with a set (s1, t1), . . . , (sk, tk) of source–sinkvertex pairs. We also call such pairs commodities. Each player is identified with onecommodity; note that different players can originate from different source vertices andtravel to different sink vertices. We use Pi to denote the si–ti paths of a network. Weconsider only networks in which Pi �= ∅ for all i, and define P = ∪k

i=1Pi . We allow thegraph G to contain parallel edges, and a vertex can participate in multiple source–sinkpairs.

We describe the routes chosen by players using a flow, which is simply a nonnegativevector indexed by the set P of source–sink paths. For a flow f and a path P ∈ Pi ,we interpret fP as the amount of traffic of commodity i that chooses the path P totravel from si to ti . Traffic is “inelastic,” in that there is a prescribed amount ri of trafficidentified with each commodity i. A flow f is feasible for a vector r if it routes all ofthe traffic: for each i ∈ {1, 2, . . . , k}, ∑

P∈PifP = ri . In particular, we do not impose

explicit edge capacities.Finally, each edge e of a network has a cost function ce : R+ → R+. We always

assume that cost functions are nonnegative, continuous, and nondecreasing. All ofthese assumptions are reasonable in applications where cost represents a quantity thatonly increases with the network congestion; delay is one natural example. When westudy the price of anarchy in Section 18.4, we also explore more severe assumptionson the network cost functions. We define a nonatomic selfish routing game, or simplya nonatomic instance, by a triple of the form (G, r, c).

Next we formalize the notion of equilibrium in nonatomic selfish routing games.Define the cost of a path P with respect to a flow f as the sum of the costs ofthe constituent edges: cP (f ) = ∑

e∈P ce(fe), where fe = ∑P∈P : e∈P fP denotes the

amount of traffic using paths that contain the edge e. Since we expect selfish traffic toattempt to minimize its cost, we arrive at the following definition.

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models and examples 461

Definition 18.1 (Nonatomic equilibrium flow) Let f be a feasible flow forthe nonatomic instance (G, r, c). The flow f is an equilibrium flow if, for everycommodity i ∈ {1, 2, . . . , k} and every pair P, P ∈ Pi of si–ti paths with fP > 0,

cP (f ) ≤ cP (f ).

In other words, all paths in use by an equilibrium flow f have minimum-possiblecost (given their source, sink, and the congestion caused by f ). In particular, all pathsof a given commodity used by an equilibrium flow have equal cost. Section 18.3.1proves that every nonatomic instance admits at least one equilibrium flow, and that allequilibrium flows of a nonatomic instance have equal cost.

In Pigou’s example, routing all of the traffic on the second link defines an equilibriumflow; only one path carries flow, and the only alternative has equal cost. Splitting thetraffic equally between the two links defines a flow that is not an equilibrium flow;the first link carries a strictly positive amount of traffic and its cost is 1, but there is astrictly cheaper alternative (the second link, with cost 1/2).

Remark 18.2 Our description of nonatomic selfish routing games and theirequilibria does not parallel that of simultaneous-move games in Chapter 1. Forexample, we have not explicitly defined the set of players. While more generaltypes of nonatomic games are frequently defined explicitly in terms of playersets, strategy profiles, and player payoff functions, selfish routing games possessspecial structure. In particular, the cost incurred by a player depends only on itspath and the amount of flow on the edges of its path, rather than on the identitiesof any of the players. Games of this type are often called congestion games.Because of this structure, it is sufficient and simpler to work directly with flowsin nonatomic selfish routing games.

When we quantify the inefficiency of equilibrium flows in Section 18.4, we consideronly the utilitarian objective of minimizing the total cost incurred by traffic. (Otherobjectives have been studied; see Section 18.6.) Precisely, since the cost incurred by aplayer choosing the path P in the flow f is cP (f ), and fP denotes the amount of trafficchoosing the path P , we define the cost of a flow f as

C(f ) =∑P∈P

cP (f )fP . (18.1)

Expanding cP (f ) as∑

e∈P ce(fe) and reversing the order of summation in (18.1) givesa useful alternative definition of the cost of a flow:

C(f ) =∑e∈E

ce(fe)fe. (18.2)

For an instance (G, r, c), we call a feasible flow optimal if it minimizes the cost overall feasible flows.

As in Chapter 17, the price of anarchy of a nonatomic selfish routing game, withrespect to this objective, is the ratio between the cost of an equilibrium flow and that ofan optimal flow. We can use the cost of an arbitrary equilibrium flow in lieu of that ofa worst equilibrium flow (cf. Chapter 17), since all equilibrium flows of a nonatomic

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462 routing games

t

c(x) = xp

s

c(x) = 1

Figure 18.1. A nonlinear variant of Pigou’s example (Example 18.3).

instance have equal cost (Section 18.3.1). In Pigou’s example, the equilibrium flowroutes all of the traffic on the second link and has cost 1. As we will see in Section 18.3.1,the optimal flow splits the traffic equally between the two links and has cost 3/4. Theprice of anarchy in Pigou’s example is therefore 4/3.

We conclude this section with two more important examples of nonatomic selfishrouting networks.

Example 18.3 (Nonlinear Pigou’s example) The inefficiency of the equilib-rium flow in Pigou’s example can be amplified with a seemingly minor modifica-tion to the network. Suppose that we replace the previously linear cost functionc(x) = x on the lower edge with the highly nonlinear one c(x) = xp for p large(Figure 18.1). As in Pigou’s example, the cost of the unique equilibrium flowis 1. The optimal flow routes a small ε fraction of the traffic on the upper edgeand has cost ε + (1 − ε)p+1, where ε tends to 0 as p tends to infinity. Precisely,Section 18.3.1 shows that ε = 1 − (p + 1)−1/p. As p tends to infinity, the costof the optimal flow approaches 0 and the price of anarchy grows without bound.Exercise 18.1 shows that this rate of growth is roughly p/ ln p as p → ∞.

While the price of anarchy in our final example is no larger than in Pigou’s example,it is arguably a more shocking display of the inefficiency of equilibria in selfish routingnetworks.

Example 18.4 (Braess’s Paradox) Consider the four-node network shown inFigure 18.2(a). There are two disjoint routes from s to t , each with combined cost1 + x, where x is the amount of traffic that uses the route. Assume that there isone unit of traffic. In the equilibrium flow, the traffic is split evenly between thetwo routes, and all of the traffic experiences 3/2 units of cost.

Now suppose that, in an effort to decrease the cost encountered by the traffic,we build a zero-cost edge connecting the midpoints of the two existing routes.The new network is shown in Figure 18.2(b). What is the new equilibrium flow?

The previous equilibrium flow does not persist in the new network: the cost ofthe new route s → v → w → t is never worse than that along the two originalpaths, and it is strictly less whenever some traffic fails to use it. As a consequence,the unique equilibrium flow routes all of the traffic on the new route. Because ofthe ensuing heavy congestion on the edges (s, v) and (w, t), all of the traffic now

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s t

w

v

c(x) = x c(x) = 1

c(x) = 1

(a) Initial network (b) Augmented network

c(x) = x

s t

w

v

c(x) = x

c(x) = x

c(x) = 1

c(x) = 1

c(x) = 0

Figure 18.2. Braess’s Paradox. The addition of an intuitively helpful edge can adversely affectall of the traffic.

experiences two units of cost. Braess’s Paradox thus shows that the intuitivelyhelpful action of adding a new zero-cost edge can increase the cost experiencedby all of the traffic!

Braess’s Paradox also has remarkable analogues in several physical systems; seeSection 18.6 for details.

The optimal flow in the second network of Example 18.4 is the same as the equi-librium flow in the first network. The price of anarchy in the second network istherefore 4/3, the same as that in Pigou’s example. This is not entirely a coincidence;in Section 18.4.1 we prove that no nonatomic instance with cost functions of the formax + b has a price of anarchy larger than 4/3.

While this chapter does not explicitly study Braess’s Paradox, we obtain bounds onthe worst-case severity of the paradox as a consequence of our results on the price ofanarchy (Remark 18.22).

18.2.2 Atomic Selfish Routing

An atomic selfish routing game or atomic instance is defined by the same ingredients as anonatomic one: a directed graph G = (V, E), k source–sink pairs (s1, t1), . . . , (sk, tk),a positive amount ri of traffic for each pair (si, ti), and a nonnegative, continuous,nondecreasing cost function ce : R+ → R+ for each edge e. We also denote an atomicinstance by a triple (G, r, c). The intuitive difference between a nonatomic and anatomic instance is that in the former, each commodity represents a large population ofindividuals, each of whom controls a negligible amount of traffic; in the latter, eachcommodity represents a single player who must route a significant amount of traffic ona single path.

More formally, atomic instances are finite simultaneous-move games in the senseof Chapter 1. There are k players, one for each source–sink pair. Different playerscan have identical source–sink pairs. The strategy set of player i is the set Pi of si–tipaths, and if player i chooses the path P , then it routes its ri units of traffic on P . Aflow is now a nonnegative vector indexed by players and paths, with f

(i)P denoting the

amount of traffic that player i routes on the si–ti path P . A flow f is feasible for an

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464 routing games

atomic instance if it corresponds to a strategy profile: for each player i, f(i)P equals ri

for exactly one si–ti path and equals 0 for all other paths. The cost cP (f ) of a path P

with respect to a flow f and the cost C(f ) of a flow f are defined as in Section 18.2.1.An equilibrium flow of an atomic selfish routing game is a feasible flow such that

no player can strictly decrease its cost by choosing a different path for its traffic.

Definition 18.5 (Atomic equilibrium flow) Let f be a feasible flow for theatomic instance (G, r, c). The flow f is an equilibrium flow if, for every playeri ∈ {1, 2, . . . , k} and every pair P, P ∈ Pi of si–ti paths with f

(i)P > 0,

cP (f ) ≤ cP (f ),

where f is the flow identical to f except that f(i)P = 0 and f

(i)P

= ri .

We have defined equilibrium flows to correspond to pure-strategy Nash equilibria (seeChapter 1). Flows corresponding to mixed-strategy Nash equilibria have also beenstudied (see Section 18.6), but we will not consider them in this chapter.

While the definitions of nonatomic and atomic instances are very similar, the twomodels are technically quite different. The next example illustrates two of these differ-ences. First, different equilibrium flows of an atomic instance can have different costs;as claimed in Section 18.2.1 and proved in Section 18.3.1, all equilibrium flows of anonatomic instance have equal cost. Second, the price of anarchy in atomic instancescan be larger than in their nonatomic counterparts. The following atomic instance hasaffine cost functions – of the form ax + b – and its price of anarchy is 5/2; in everynonatomic instance with affine cost functions, the price of anarchy is at most 4/3(Section 18.4.1). We call this the AAE example, after the initials of its discoverers (seeSection 18.6).

Example 18.6 (AAE example) Consider the bidirected triangle network shownin Figure 18.3. We assume that there are four players, each of whom needs to routeone unit of traffic. The first two have source u and sinks v and w, respectively;the third has source v and sink w; and the fourth has source w and sink v. Eachplayer has two strategies, a one-hop path and a two-hop path. In the optimal flow,all players route on their one-hop paths, and the cost of this flow is 4. This flow isalso an equilibrium flow. On the other hand, if all players route on their two-hoppaths, then we obtain a second equilibrium flow. Since the first two players eachincur three units of cost and the last two players each incur two units of cost, thisequilibrium flow has a cost of 10. The price of anarchy of this instance is therefore10/4 = 2.5.

Exercise 18.2 explores variants of the AAE example.Next we study the even more basic issue of the existence of equilibrium flows.

Recall that equilibrium flows for atomic instances correspond to pure-strategy Nashequilibria, which do not always exist in arbitrary finite games (see Chapter 1). Dothey always exist in atomic selfish routing games? Our second example answers thisquestion in the negative.

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s1u

w

0

x

x

xx0

s2

t1

t2 t3 s4

s3v t4

Figure 18.3. The AAE example (Example 18.6). In atomic instances with affine cost functions,different equilibrium flows can have different costs, and the price of anarchy can be as largeas 5/2.

Example 18.7 (Nonexistence in weighted atomic instances) Consider the net-work shown in Figure 18.4. Extend this network to an atomic selfish routing gameby adding two players, both with source s and sink t , with traffic amounts r1 = 1and r2 = 2.

We claim that there is no equilibrium flow in this atomic instance. To provethis, let P1, P2, P3, and P4 denote the paths s → t , s → v → t , s → w → t ,and s → v → w → t , respectively. The following four statements then imply theclaim.

(1) If player 2 takes path P1 or P2, then the unique response by player 1 that minimizesits cost is the path P4.

(2) If player 2 takes path P3 or P4, then the unique best response by player 1 is thepath P1.

(3) If player 1 takes the path P4, then the unique best response by player 2 is thepath P3.

(4) If player 1 takes the path P1, then the unique best response by player 2 is thepath P2.

We leave verification of (1)–(4) to the reader.

On the other hand, Section 18.3.2 proves that every atomic instance in which allplayers route the same amount of traffic admits at least one equilibrium flow. We call

s t

w

v

47x

x2 + 443x2

6x2x + 33 13x

Figure 18.4. An atomic instance with no equilibrium flow (Example 18.7).

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466 routing games

instances of this type unweighted. Example 18.6 is an unweighted instance, whileExample 18.7 is not.

18.3 Existence, Uniqueness, and Potential Functions

This section collects existence and uniqueness results about equilibrium flows innonatomic and atomic selfish routing games. We also introduce the potential func-tion method, a fundamental proof technique.

18.3.1 Nonatomic Selfish Routing: Existence and Uniqueness

Our next goal is to show that in nonatomic selfish routing games, equilibrium flowsalways exist and are essentially unique. By “essentially unique,” we mean that allequilibrium flows of a nonatomic instance have the same cost. In particular, the priceof stability (Section 17.1) and the price of anarchy coincide in every nonatomic instance.Formally, our aim is to prove the following theorem.

Theorem 18.8 (Existence and uniqueness of equilibrium flows) Let (G, r, c)be a nonatomic instance.

(a) The instance (G, r, c) admits at least one equilibrium flow.

(b) If f and f are equilibrium flows for (G, r, c), then ce(fe) = ce(f e) for everyedge e.

Part (b) of the theorem and Definition 18.1 easily imply that two equilibrium flows ofa nonatomic instance have equal cost.

We prove Theorem 18.8 with the potential function method. The idea of this methodis to exhibit a real-valued “potential function,” defined on the outcomes of a game, suchthat the equilibria of the game are precisely the outcomes that optimize the potentialfunction. Potential functions are useful because they enable the application of optimiza-tion techniques to the study of equilibria. When a game admits a potential function, thereare typically consequences for the existence, uniqueness, and inefficiency of equilibria.

To motivate the potential functions corresponding to nonatomic selfish routinggames, we present a characterization of optimal flows in such games. To state this char-acterization cleanly, we assume that for every edge e of the given nonatomic instance,the function x · ce(x) is continuously differentiable and convex. Note that x · ce(x) isthe contribution to the social cost function (18.2) by traffic on the edge e. Let c∗

e (x) =(x · ce(x))′ = ce(x) + x · c′

e(x) denote the marginal cost function for the edge e. Forexample, if c(x) denotes the cost function c(x) = axp for some a, p ≥ 0, then the cor-responding marginal cost function is c∗(x) = (p + 1)axp. Let c∗

P (f ) = ∑e∈P c∗

e (f )denote the sum of the marginal costs of the edges in the path P with respect to theflow f . The characterization follows.

Proposition 18.9 (Characterization of optimal flows) Let (G, r, c) be a non-atomic instance such that, for every edge e, the function x · ce(x) is convexand continuously differentiable. Let c∗

e denote the marginal cost function of the

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existence, uniqueness, and potential functions 467

edge e. Then f ∗ is an optimal flow for (G, r, c) if and only if, for every commodityi ∈ {1, 2, . . . , k} and every pair P, P ∈ Pi of si–ti paths with f ∗

P > 0,

c∗P (f ∗) ≤ c∗

P(f ∗).

Proposition 18.9 follows immediately from the first-order conditions of a convex opti-mization problem with nonnegativity constraints. We omit the details and focus insteadon how the proposition leads to a potential function for equilibrium flows in nonatomicinstances, and on the implications of this potential function for the existence anduniqueness of equilibrium flows.

Definition 18.1 and Proposition 18.9 immediately imply that equilibrium flows andoptimal flows are the same thing, just with respect to different sets of cost functions.

Corollary 18.10 (Equivalence of equilibrium and optimal flows) Let (G, r, c)be a nonatomic instance such that, for every edge e, the function x · ce(x) is convexand continuously differentiable. Let c∗

e denote the marginal cost function of theedge e. Then f ∗ is an optimal flow for (G, r, c) if and only if it is an equilibriumflow for (G, r, c∗).

For instance, in Pigou’s example (Example 17.1), the marginal cost functions of thetwo edges are c∗(x) = 1 and c∗(x) = 2x. The equilibrium flow with respect to themarginal cost functions splits the traffic equally between the two links, equalizing theirmarginal costs at 1; by Corollary 18.10, this flow is optimal in the original network. Inthe nonlinear variant of Pigou’s example (Example 18.3), the marginal cost functionsare c∗(x) = 1 and c∗(x) = (p + 1)xp; the optimal flow therefore routes (p + 1)−1/p

units of traffic on the second link and the rest on the first. In Braess’s Paradox with thezero-cost edge added (Example 18.4 and Figure 18.2(b)), routing half of the traffic oneach of the paths s → v → t and s → w → t equalizes the marginal costs of all threepaths at 2, and therefore provides an optimal flow for the original instance.

To construct a potential function for equilibrium flows, we need to “invert” Corol-lary 18.10: of what function do equilibrium flows arise as the global minima? Theanswer is simple: to recover Definition 18.1 as an optimality condition, we seek afunction he(x) for each edge e – playing the previous role of x · ce(x) – such thath′

e(x) = ce(x). Setting he(x) = ∫ x

0 ce(y) dy for each edge e thus yields the desired po-tential function. Moreover, since ce is continuous and nondecreasing for every edge e,every function he is both continuously differentiable and convex.

Precisely, call

�(f ) =∑e∈E

∫ fe

0ce(x) dx (18.3)

the potential function of a nonatomic instance (G, r, c). Invoking Proposition 18.9,with each function x · ce(x) replaced by he(x) = ∫ x

0 c(y) dy, yields the same conditionas in Definition 18.1; we have therefore characterized equilibrium flows as the globalminimizers of the potential function �.

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468 routing games

Proposition 18.11 (Potential function for equilibrium flows) Let (G, r, c) bea nonatomic instance. A flow feasible for (G, r, c) is an equilibrium flow if andonly if it is a global minimum of the corresponding potential function � givenin (18.3).

Theorem 18.8 now follows from Proposition 18.11 and routine calculus.

Proof of Theorem 18.8 We first note that, by definition, the set of feasibleflows of (G, r, c) can be identified with a compact (i.e., closed and bounded) sub-set of |P|-dimensional Euclidean space. Since edge cost functions are continuous,the potential function is a continuous function on this set. By Weierstrass’s Theo-rem from elementary mathematical analysis, the potential function � achieves aminimum value on this set. By Proposition 18.11, every point at which � attainsits minimum corresponds to an equilibrium flow of (G, r, c).

For part (b), recall that each cost function is nondecreasing, and hence eachsummand on the right-hand side of (18.3) is convex. Hence, the potential func-tion � is a convex function.

Now suppose that f and f are equilibrium flows for (G, r, c). By Proposi-tion 18.11, both f and f minimize the potential function �. We consider allconvex combinations of f and f – that is, all vectors of the form λf + (1 − λ)ffor λ ∈ [0, 1]. All of these vectors are feasible flows. Since � is a convex function,a chord between two points on its graph cannot pass below its graph. In algebraicterms, we have

�(λf + (1 − λ)f ) ≤ λ�(f ) + (1 − λ)�(f ) (18.4)

for every λ ∈ [0, 1]. Since both f and f are global minima of �, the inequal-ity (18.4) must hold with equality for all of their convex combinations. Sinceevery summand of � is convex, this can occur only if every summand

∫ x

0 ce(y) dy

is linear between the values fe and f e. In turn, this implies that every cost functionce is constant between fe and f e.

18.3.2 Atomic Selfish Routing: Existence

We now consider equilibrium flows in atomic instances. The AAE example (Exam-ple 18.6) suggests that no interesting uniqueness results are possible in such instances,so we focus instead on the existence of equilibrium flows. Similarly, Example 18.7demonstrates that a general atomic instance need not admit an equilibrium flow. Thereare two approaches to circumventing this counterexample. The first, taken in thissection, is to place additional restrictions on atomic instances so that equilibriumflows are guaranteed to exist. The second approach, discussed in Remark 18.26,is to relax the equilibrium concept so that an equilibrium exists in every atomicinstance.

The key result in this section is the following theorem, which establishes the exis-tence of equilibrium flows in atomic instances in which all players control the sameamount of traffic.

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existence, uniqueness, and potential functions 469

Theorem 18.12 (Equilibrium flows in unweighted atomic instances) Let(G, r, c) be an atomic instance in which every traffic amount ri is equal to acommon positive value R. Then (G, r, c) admits at least one equilibrium flow.

proof We obtain Theorem 18.12 by discretizing the potential function (18.3)for nonatomic instances and the proof of Theorem 18.8(a). Assume for simplicitythat R = 1. Set

�a(f ) =∑e∈E

fe∑i=1

ce(i) (18.5)

for every feasible flow f . Note that �a is the same as the previous potentialfunction � for nonatomic instances, except that the integral

∫ fe

0 c(x) dx has been

replaced by the sum∑fe

i=1 ce(i).Since the atomic instance (G, r, c) has a finite number of players, and each of

these has a finite number of strategies, there are only a finite number of possibleflows. One of these, call it f , is a global minimum of the potential function �a .We claim that f is an equilibrium flow for (G, r, c). To prove it, assume forcontradiction that in f , the player i could strictly decrease its cost by deviatingfrom the path P to the path P , yielding the new flow f . In other words, we assumethat

0 > cP (f ) − cP (f ) =∑

e∈P \Pce(fe + 1) −

∑e∈P \P

ce(fe). (18.6)

On the other hand, consider the impact of player i’s deviation on the potentialfunction �a: for edges in P \ P , the corresponding sum in (18.5) acquires theextra term ce(fe + 1); for edges in P \ P , the corresponding sum sheds the termce(fe); and for edges of P ∩ P , the corresponding sum remains the same. Thus,�a(f ) − �a(f ) is precisely the third expression of (18.6). Since this expressionis negative, the potential function value of f is strictly less than that of f , whichcontradicts our choice of f .

Remark 18.13 The proof of Theorem 18.12 establishes a remarkable propertyof the potential function �a: it “tracks” the change in cost experienced by adeviating player. More formally, for every flow, every player, and every deviationby a player, the change in the player’s cost is identical to the change in thepotential function. This property has consequences beyond the existence resultof Theorem 18.12. For example, it implies that “best-response dynamics” areguaranteed to converge to an equilibrium flow. See Chapter 19 for further details.

Remark 18.14 The proof of Theorem 18.12 did not use any assumptions aboutthe edge cost functions. In particular, it is also valid when cost functions arenot nondecreasing. This property will be crucial for some of the network designgames studied in Chapter 19, which can be viewed as atomic selfish routing gameswith decreasing cost functions.

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The next theorem guarantees the existence of equilibrium flows under a differentrestriction – affine cost functions. (Recall that a cost function ce(x) is affine if it has theform aex + be; we always assume that ae, be ≥ 0.)

Theorem 18.15 (Equilibrium flows with affine cost functions) Let (G, r, c)be an atomic instance with affine cost functions. Then (G, r, c) admits at least oneequilibrium flow.

The proof of Theorem 18.15 follows the same outline as that of Theorem 18.12, anduses a variant of the potential function method. See Exercise 18.4 for further details.

18.4 The Price of Anarchy of Selfish Routing

18.4.1 Nonatomic Selfish Routing: The Price of Anarchy

This section gives an essentially complete analysis of the price of anarchy in nonatomicselfish routing games. As we know from the nonlinear variant of Pigou’s example(Example 18.3), the price of anarchy depends on “nonlinearity” of the network costfunctions. Our goal is to show that it depends on nothing else – not the network size, thenetwork structure, nor the number of commodities. More precisely, we show that forevery conceivable restriction on the cost functions of a network, the price of anarchy ismaximized (over all multicommodity networks) by the network that best “simulates”Pigou’s example and its nonlinear variants.

As an aside, we note that the potential function characterization of nonatomicequilibrium flows (Proposition 18.11) already gives a good, but not optimal, upperbound on the price of anarchy. The intuitive explanation is simple: if equilibriumflows exactly optimize a potential function (18.3) that is a good approximation of theobjective function (18.2), then they should also be approximately optimal.

Theorem 18.16 (Potential function upper bound) Let (G, r, c) be a nonatomicinstance, and suppose that x · ce(x) ≤ γ · ∫ x

0 ce(y) dy for all e ∈ E and x ≥ 0.Then the price of anarchy of (G, r, c) is at most γ .

proof Let f and f ∗ be equilibrium and optimal flows for (G, r, c), respectively.Since cost functions are nondecreasing, the cost of a flow (18.2) is always at leastits potential function value (18.3). The hypothesis ensures that the cost of a flowis at most γ times its potential function value. The theorem follows by writing

C(f ) ≤ γ · �(f ) ≤ γ · �(f ∗) ≤ γ · C(f ∗),

with the second inequality following from Proposition 18.11.

Theorem 18.16 implies that the price of anarchy of selfish routing is large onlyin networks with “highly nonlinear” cost functions. For example, if ce is a polyno-mial function with degree at most p and nonnegative coefficients, then x · ce(x) ≤(p + 1)

∫ x

0 ce(y) dy for all x ≥ 0. Theorem 18.16 then shows that the price of anarchyin nonatomic instances with such cost functions is at most linear in p.

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the price of anarchy of selfish routing 471

Corollary 18.17 (Potential function bound for polynomials) If (G, r, c) is anonatomic instance with cost functions that polynomials with nonnegative coef-ficients and degree at most p, then the price of anarchy of (G, r, c) is at mostp + 1.

This upper bound is nearly matched by Example 18.3, although the upper and lowerbounds differ by roughly a ln p multiplicative factor (Exercise 18.1). We close thisgap using a different and important proof technique, which is driven by variationalinequalities.

We first formalize a natural lower bound on the price of anarchy based on “Pigou-likeexamples.”

Definition 18.18 (Pigou bound) Let C be a nonempty set of cost functions. ThePigou bound α(C) for C is

α(C) = supc∈C

supx,r≥0

r · c(r)

x · c(x) + (r − x)c(r), (18.7)

with the understanding that 0/0 = 1.

The point of the Pigou bound is that it lower bounds the price of anarchy in instanceswith cost functions in C.

Proposition 18.19 Let C be a set of cost functions that contains all of theconstant cost functions. Then the price of anarchy in nonatomic instances withcost functions in C can be arbitrarily close to α(C).

proof Fix a choice of c ∈ C and x, r ≥ 0. We can complete the proof byexhibiting a selfish routing network with cost functions in C and price of anarchyat least c(r)r/[c(x)x + (r − x)c(r)]. Since c is nondecreasing, this expression isat most 1 if x ≥ r; we can therefore assume that x < r .

Let G be a two-vertex, two-edge network as in Figure 18.1. Give the loweredge the cost function c1(y) = c(y) and the upper edge the constant cost functionc2(y) = c(r). By assumption, both of these cost functions lie in C. Set the trafficrate to r . Routing all of the traffic on the lower edge yields an equilibrium flowwith cost c(r)r . Routing x units of traffic on the lower edge and r − x units oftraffic on the upper edge gives a feasible flow with cost [c(x)x + (r − x)c(r)].The price of anarchy in this instance is thus at least c(r)r/[c(x)x + (r − x)c(r)],as desired.

While Proposition 18.19 assumes that the set C includes all of the constant cost func-tions, its conclusion holds whenever C is inhomogeneous in the sense that c(0) > 0 forsome c ∈ C (Exercise 18.5).

We next show that, even though the Pigou bound is based only on Pigou-likeexamples, it is also an upper bound on the price of anarchy in general multicommodityflow networks. The proof requires the following variational inequality characterizationof equilibrium flows.

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Proposition 18.20 (Variational inequality characterization) Let f be a fea-sible flow for the nonatomic instance (G, r, c). The flow f is an equilibrium flowif and only if ∑

e∈E

ce(fe)fe ≤∑e∈E

ce(fe)f ∗e

for every flow f ∗ feasible for (G, r, c).

proof Fix f and define the function Hf on the set of feasible flows by

Hf (f ∗) =k∑

i=1

∑P∈Pi

cP (f )f ∗P =

∑e∈E

ce(fe)f ∗e ;

the same reversal of sums used to prove the equivalence of (18.1) and (18.2)shows that these two definitions of Hf (f ∗) agree. The value Hf (f ∗) denotes thecost of a flow f ∗ after the cost function of each edge e has been changed to theconstant function everywhere equal to ce(fe). By the second definition of Hf , theproposition is equivalent to the assertion that a flow f is an equilibrium flow ifand only if it minimizes Hf (·) over all feasible flows.

Examining the first definition of Hf shows that a flow f ∗ minimizes Hf ifand only if, for every commodity i, f ∗

P > 0 only for paths P that minimize cP (f )over all si–ti paths. Since the flow f satisfies this condition if and only if it is anequilibrium flow, the proof is complete.

We now show that the Pigou bound is tight.

Theorem 18.21 (Tightness of the Pigou bound) Let C be a set of cost func-tions and α(C) the Pigou bound for C. If (G, r, c) is a nonatomic instance withcost functions in C, then the price of anarchy of (G, r, c) is at most α(C).

proof Let f ∗ and f be optimal and equilibrium flows, respectively, for anonatomic instance (G, r, c) with cost functions in the set C. The theorem followsby writing

C(f ∗) =∑e∈E

ce(f ∗e )f ∗

e

≥ 1

α(C)

∑e∈E

ce(fe)fe +∑e∈E

(f ∗e − fe)ce(fe)

≥ C(f )

α(C),

where the first inequality follows from Definition 18.18, applied to each edgee with x = f ∗

e and r = fe, and the second inequality follows from Proposi-tion 18.20.

Proposition 18.19 and Theorem 18.21 show that, for essentially every fixed restric-tion on the allowable cost functions, the price of anarchy is maximized by Pigou-like

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examples. Determining the largest-possible price of anarchy in Pigou-like examples(i.e., the Pigou bound) is a tractable problem in many cases. For example, it is pre-cisely 4/3 when C is the set of affine cost functions (Exercise 18.6), and more generallyis [1 − p · (p + 1)−(p+1)/p]−1 ≈ p/ ln p when C is the set of polynomials with degreeat most p and nonnegative coefficients. In these cases, the maximum price of anarchy(among all multicommodity instances) is achieved by the instances in Examples 17.1and 18.3. The Pigou bound is also known for several other classes of cost functions;see Section 18.6 for references.

Remark 18.22 (Bounds on Braess’s Paradox) Braess’s Paradox (Example18.4) shows that adding edges to a network can increase the cost of its equi-librium flow. Since the equilibrium flow in the original network is a candidate forthe optimal flow in the second network, the ratio between the costs of the new andoriginal equilibrium flows is a lower bound on the price of anarchy in the latternetwork.

On the other hand, Theorem 18.21 and Exercise 18.6 show that the price ofanarchy is at most 4/3 in every network with affine cost functions. Thus, addingedges to a network with affine cost functions cannot increase the cost of itsequilibrium flow by more than a 4/3 factor. Example 18.4 is therefore a worst-case manifestation of Braess’s Paradox in networks with affine cost functions.Similar bounds also apply to the physical analogues of Braess’s Paradox that aredescribed in Section 18.6.

18.4.2 Atomic Selfish Routing: The Price of Anarchy

We now consider atomic selfish routing games. We again obtain tight bounds on theprice of anarchy, at least for polynomial cost functions, but the discrete nature of atomicinstances complicates the analysis.

We first note that the potential function method, which gave nontrivial bounds on theprice of anarchy for nonatomic instances (Theorem 18.16), cannot be used for atomicinstances. The difficulty stems from the non-uniqueness of equilibrium flows in atomicinstances (Example 18.6). Recall that a bound on the price of anarchy is a guaranteethat all equilibrium flows of an instance are nearly optimal. Reviewing the proof ofTheorem 18.16, we observe that the potential function method argues about only oneequilibrium flow – the one with minimum potential function value. As a result, thepotential function method is directly useful only for bounding the price of stabilityrather than the price of anarchy. While these two quantities coincide in nonatomicselfish routing games, they are generally different in atomic ones. (See Section 18.6for results on the price of stability in atomic selfish routing games.)

We instead rely on proof techniques that are partially inspired by the variationalinequality of Proposition 18.20. This inequality expresses the fact that equilibriumflows route all traffic on shortest paths, with respect to the induced edge costs. We derivea similar, if more complicated, condition for atomic instances. To keep the proofs astransparent as possible, we focus on atomic instances with affine cost functions. Recallfrom Theorem 18.15 that every such instance admits at least one equilibrium flow. The

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474 routing games

analysis can also be extended to other cost functions and other equilibrium concepts;see Remark 18.26 and Section 18.6 for more details.

Our goal is the following theorem.

Theorem 18.23 (The price of anarchy in affine atomic instances) If (G, r, c)is an atomic instance with affine cost functions, then the price of anarchy of(G, r, c) is at most (3 + √

5)/2 ≈ 2.618.

A variant of the AAE example (Example 18.6) shows that the upper bound in Theo-rem 18.23 is the best possible if different players can control different amounts of flow(Exercise 18.2(a)). If all of the players control the same amount of flow, then a variantof the following proof gives an improved upper bound of 5/2, which matches the lowerbound furnished by the AAE example (Exercise 18.7).

We build up to Theorem 18.23 in a sequence of steps. We begin with a lemma thatfollows immediately from the definition of an equilibrium flow.

Lemma 18.24 (Equilibrium condition) Let (G, r, c) be an atomic instance inwhich each edge e has an affine cost function ce(x) = aex + be with ae, be ≥ 0.Let f and f ∗ be equilibrium and optimal flows, respectively, for (G, r, c). Letplayer i use the path Pi in f and the path P ∗

i in f ∗. Then∑e∈Pi

[aefe + be] ≤∑e∈P ∗

i

[ae(fe + ri) + be]. (18.8)

Our second step is to combine the inequalities of Lemma 18.24 – one per player –to relate the cost of an arbitrary equilibrium flow to that of an optimal flow.

Lemma 18.25 (Equilibrium inequality) With the same assumptions and nota-tion as in Lemma 18.24,

C(f ) ≤ C(f ∗) +∑e∈E

aefef∗e . (18.9)

proof For each player i, multiply the inequality (18.8) by ri . Summing up theresulting k inequalities, we obtain

C(f ) ≤k∑

i=1

ri

⎛⎝∑

e∈P ∗i

ae(fe + ri) + be

⎞⎠

≤k∑

i=1

ri

⎛⎝∑

e∈P ∗i

ae(fe + f ∗e ) + be

⎞⎠

=∑e∈E

[ae(fe + f ∗

e ) + be

]f ∗

e ,

where the equality follows by reversing the order of summation. Since the finalexpression equals the right-hand side of (18.9), the proof is complete.

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To complete the proof of Theorem 18.23, we upper bound the magnitude of the“error term” in (18.9) relative to the costs of the equilibrium and optimal flows.

Proof of Theorem 18.23 Let f and f ∗ denote equilibrium and optimalflows, respectively, for the atomic instance (G, r, c). Assume that edge e hasthe cost function ce(x) = aex + be for ae, be ≥ 0. Apply the Cauchy–SchwarzInequality to the vectors {√aefe}e∈E and {√aef

∗e }e∈E to obtain

∑e∈E

aefef∗e ≤

√∑e∈E

aef 2e ·

√∑e∈E

ae(f ∗e )2 ≤

√C(f ) ·

√C(f ∗).

Combining this with the Equilibrium Inequality (18.9), dividing throughby C(f ∗), and rearranging gives

C(f )

C(f ∗)− 1 ≤

√C(f )

C(f ∗).

Squaring both sides and solving the corresponding quadratic inequality x2 − 3x +1 ≤ 0, we find that

C(f )

C(f ∗)≤ 3 + √

5

2≈ 2.618,

as claimed.

Theorem 18.23 can be extended to atomic instances with cost functions that arepolynomials with nonnegative coefficients and degree at most a parameter p. However,the upper bound on the price of anarchy increases with p roughly in proportion to theexponential function pp – much faster than in nonatomic instances. This exponentialdependence is not an artifact of the above proof approach, as nearly matching lowerbounds on the price of anarchy are known (Section 18.6).

Remark 18.26 Strictly speaking, the price of anarchy is not always defined ingeneral atomic instances, where equilibrium flows need not exist (Example 18.7).Nevertheless, Theorem 18.23 has been extended to atomic instances with poly-nomial cost functions in three different ways. First, when such an instance doesadmit at least one equilibrium flow, then all such flows have cost at most pO(p)

times that of an optimal flow. Second, by Nash’s Theorem (Chapters 1 and 2),every such instance admits a mixed-strategy Nash equilibrium, and the expectedcost of every such equilibrium is at most pO(p) times that of an optimal flow. Fi-nally, similar upper bounds have been proved for “sink equilibria,” an equilibriumconcept that always exists in finite games and is motivated by convergence issues.

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476 routing games

18.5 Reducing the Price of Anarchy

As we have seen, the price of anarchy can be large in both nonatomic and atomic selfishrouting games when cost functions are highly nonlinear. This motivates a question firstposed in Section 17.3: how can we design or modify a selfish routing network, withoutexplicitly imposing an optimal solution, to minimize the inefficiency of its equilibria?Can modest intervention significantly reduce the price of anarchy? We briefly discusstwo techniques for mitigating the inefficiency of selfish routing in nonatomic instances:influencing traffic with edge taxes (Subsection 18.5.1) and increasing the capacity ofthe network (Subsection 18.5.2).

18.5.1 Marginal Cost Pricing

Our first approach to reducing the price of anarchy in nonatomic selfish routing gamesis to use marginal cost taxes on the edges of the network. The idea of marginal costpricing is to charge each network user on each edge for the additional cost its presencecauses for the other users of the edge. To discuss this idea formally, we allow each edgee of a nonatomic selfish routing network to possess a nonnegative tax τe. We denote anonatomic instance (G, r, c) with edge taxes τ by (G, r, c + τ ). An equilibrium flow forsuch an instance (G, r, c + τ ) is defined as in Definition 18.1, with all traffic travelingon routes that minimize the sum of the edge costs and edge taxes. Equivalently, itis an equilibrium flow for the nonatomic instance (G, r, cτ ), where the cost functioncτe is a shifted version of the original cost function ce: cτ

e (x) = ce(x) + τe for allx ≥ 0.

The principle of marginal cost pricing asserts that for a flow f feasible for anonatomic instance (G, r, c), the tax τe assigned to the edge e should be τe = fe · c′

e(fe),where c′

e denotes the derivative of ce. (Assume for simplicity that the cost functionsare differentiable.) The term c′

e(fe) corresponds to the marginal increase in cost causedby one user of the edge, and the term fe is the amount of traffic that suffers fromthis increase. We can also interpret the marginal cost tax τe using Corollary 18.10:τe is precisely the “extra term” in the marginal cost function that is absent from theoriginal cost function. These taxes correct for the failure of selfish users to account forthe second, “altruistic” term of the marginal cost function. Formally, Corollary 18.10easily implies the following guarantee.

Theorem 18.27 Let (G, r, c) be a nonatomic instance such that, for every edgee, the function x · ce(x) is convex and continuously differentiable. Let f ∗ be anoptimal flow for (G, r, c) and let τe = f ∗

e · c′e(f ∗

e ) denote the marginal cost taxfor edge e with respect f ∗. Then f ∗ is an equilibrium flow for (G, r, c + τ ).

Marginal cost taxes thus induce an optimal flow as an equilibrium flow; in thissense, such taxes reduce the price of anarchy to 1. Theorem 18.27 also holds withweaker assumptions on the cost functions; in particular, the convexity hypothesisis not needed. For further discussion of pricing problems in routing games, seeChapter 22.

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reducing the price of anarchy 477

18.5.2 Capacity Augmentation

Our final result is a novel type of bound on the inefficiency of equilibrium flows innonatomic selfish routing games with arbitrary cost functions. This bound does notinvolve the price of anarchy, which is unbounded in such networks (Example 18.3),and instead shows that the cost of an equilibrium flow is at most that of an optimalflow that is forced to route twice as much traffic between each source–sink pair. As wewill see, this result implies that in lieu of centralized control, the inefficiency of selfishrouting can be offset by a moderate increase in link speed.

Example 18.28 Consider the nonlinear variant of Pigou’s example (Exam-ple 18.3). When there is one unit of traffic, the equilibrium flow routes all of theflow on the lower edge, while the optimal flow routes ε units of flow on the upperedge and the rest on the lower edge (where ε → 0 as p → ∞). When the amountr of traffic to be routed exceeds one, an optimal flow assigns the additional r − 1units of traffic to the upper link, incurring a cost that tends to r − 1 as p → ∞.In particular, for every p an optimal flow feasible for twice the original trafficamount (r = 2) has cost at least 1, the cost of the equilibrium flow in the originalinstance.

We now show that the upper bound stated in Example 18.28 for the nonlinear variantof Pigou’s example holds in every nonatomic instance.

Theorem 18.29 If f is an equilibrium flow for (G, r, c) and f ∗ is feasible for(G, 2r, c), then C(f ) ≤ C(f ∗).

proof Let f and f ∗ denote an equilibrium flow for (G, r, c) and a feasible flowfor (G, 2r, c), respectively. For each commodity i, let di denote the minimum costof an si–ti path with respect to the flow f . Definition 18.1 and the definition ofcost (18.1) imply that C(f ) = ∑

i ridi .The key idea is to define a set of cost functions c that satisfies two properties:

lower bounding the cost of f ∗ relative to that of f is easy with respect to c; andthe new cost functions c approximate the original ones c. Specifically, we setce(x) = max{ce(fe), ce(x)} for each edge e. Let C(·) denote the cost of a flow inthe instance (G, r, c). Note that C(f ∗) ≥ C(f ∗) while C(f ) = C(f ).

We first upper bound the amount by which the new cost C(f ∗) of f ∗ canexceed its original cost C(f ∗). For every edge e, ce(x) − ce(x) is zero for x ≥ fe

and bounded above by ce(fe) for x < fe, so x(ce(x) − ce(x)) ≤ ce(fe)fe for allx ≥ 0. Thus

C(f ∗) − C(f ∗) =∑e∈E

f ∗e (ce(f ∗

e ) − ce(f ∗e )) ≤

∑e∈E

ce(fe)fe = C(f ). (18.10)

In other words, evaluating f ∗ with cost functions c, rather than c, increases itscost by at most an additive C(f ) factor.

Now we lower bound C(f ∗). By construction, the modified cost ce(·) of an edgee is always at least ce(fe), so the modified cost cP (·) of a path P ∈ Pi is always at

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478 routing games

least cP (f ), which in turn is at least di . The modified cost C(f ∗) therefore equals

∑P∈P

cP (f ∗)f ∗P ≥

k∑i=1

∑P∈Pi

dif∗P =

k∑i=1

2ridi = 2C(f ). (18.11)

The theorem now follows immediately from inequalities (18.10) and (18.11).

Another interpretation of Theorem 18.29 is that the benefit of centralized control isequaled or exceeded by the benefit of a sufficient improvement in link technology.

Corollary 18.30 Let (G, r, c) be an instance and define the modified cost func-tion ce by ce(x) = ce(x/2)/2 for each edge e. Let f be an equilibrium flow for(G, r, c) with cost C(f ), and f ∗ a feasible flow for (G, r, c) with cost C(f ∗). ThenC(f ) ≤ C(f ∗).

Simple calculations show that Theorem 18.29 and Corollary 18.30 are equivalent; seeExercise 18.8(a).

Corollary 18.30 takes on a particularly nice form in instances in which all costfunctions are M/M/1 delay functions. Such a cost function has the form ce(x) = (ue −x)−1, where ue can be interpreted as an edge capacity or a queue service rate; thefunction is defined to be +∞ when x ≥ ue. (Rigorously allowing infinite costs inthis selfish routing model requires some care; we ignore these issues in this chapter.)In this case, the modified function ce of Corollary 18.30 is ce(x) = 1/2(ue − x/2) =1/(2ue − x). Corollary 18.30 thus suggests the following design principle for selfishrouting networks with M/M/1 delay functions: to outperform optimal routing, justdouble the capacity of every edge.

18.6 Notes

18.6.1 Nonatomic Selfish Routing

Nonatomic selfish routing was first studied in the context of transportation networks.Pigou (1920) informally discussed Pigou’s example in his 1920 book, The Economicsof Welfare, in order to illustrate the inefficiency of equilibria. He also anticipatedthe principle of marginal cost pricing discussed in Theorem 18.27; indeed, marginalcost taxes are sometimes called Pigouvian taxes. The model was first formally de-fined by Wardrop (1952). For this reason, equilibrium flows in nonatomic selfishrouting games are often called Wardrop equilibria. We use the term “equilibriumflow” so that the terminology for nonatomic and atomic selfish routing games is thesame.

Beckmann et al. (1956) proved a number of fundamental results for the nonatomicmodel. Theorem 18.8, Proposition 18.9, Corollary 18.10, Proposition 18.11, andTheorem 18.27 were first proved in Beckmann et al. (1956), via proofs essentiallyidentical to the ones given here. Details on first-order conditions for convex pro-gramming problems can be found in Bertsekas (1999, Chapter 2). Schmeidler (1973)founded the theory of general noncooperative nonatomic games.

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notes 479

Two decades after nonatomic selfish routing games were first defined, researchersbegan to use them to model the routing of data through communication networks.Nonatomic selfish routing is immediately relevant for networks that employ so-calledsource routing, meaning that each sender is responsible for selecting a full path of linksto the receiver. Assuming that senders seek paths of minimum cost, senders of data insuch networks correspond to the users of a selfish routing network.

In large networks such as the Internet, distributed shortest-path routing is typicallyused instead of source routing. In distributed shortest-path routing, each link is givena positive length, and data are forwarded along a path of minimum total length to itsdestination. Shortest-path routing leaves a key parameter unspecified: the length ofeach edge. A direct correspondence between selfish routing and shortest-path routingexists if and only if the edge cost functions coincide with the lengths used to defineshortest paths. In other words, when an x fraction of the overall network traffic isusing an edge with cost function c(·), then the corresponding shortest-path routingalgorithm should define the length of the edge as the number c(x). If the cost functionc is nonconstant, then this is a congestion-dependent definition of the edge length.In this case, shortest-path routing will route traffic exactly as if it is a network withselfish routing (or source routing). For details on this equivalence, see the textbook byBertsekas and Tsitsiklis (1989). See Qiu et al. (2003), for example, for a more recentpaper that studies selfish routing from a computer networking perspective.

Braess’s Paradox was discovered by Braess (1968). The variant in Example 18.4 wasnoted by L. Schulman (personal communication, October 1999). For surveys on thelarge literature inspired by Braess’s Paradox, see Roughgarden (2006) and D. Braess’shome page (Braess, 2007).

Cohen and Horowitz (1991) noted that Braess’s Paradox has startling analogues inphysical systems. For instance, Example 18.4 can be simulated in the following systemof strings and springs. One end of a spring is attached to a fixed support, and the otherend to a very short string. A second identical spring is hung from the free end of thestring and carries a heavy weight. Finally, strings are connected, with very little slack,from the support to the upper end of the second spring and from the lower end of thefirst spring to the weight. Assuming that the springs are ideally elastic, the stretchedlength of a spring is a linear function of the force applied to it. We can therefore viewthe network of strings and springs as a selfish routing game, where force corresponds totraffic and physical distance corresponds to cost. Remarkably, severing the very shorttaut string causes the weight to levitate away from the ground! The rise in the weight isthe same as the improvement in the equilibrium flow obtained by deleting the zero-costedge of Figure 18.2(b) to recover the network of Figure 18.2(a).

The price of anarchy in nonatomic selfish routing games was first studied byRoughgarden and Tardos (2002). The nonlinear variant of Pigou’s example (Exam-ple 18.3) is from Roughgarden and Tardos (2002), as is Theorem 18.16. Roughgardenand Tardos (2002) also proved the special case of Theorem 18.21 for networks withaffine cost functions (where the price of anarchy is at most 4/3). Roughgarden (2003)introduced the Pigou bound and proved Theorem 18.21 under the same convexityhypothesis used in Theorem 18.9. The solution to Exercise 18.5 can also be foundin Roughgarden (2003). A. Ronen (personal communication, March 2002) suggestedusing the variational inequality in Proposition 18.20, which was first proved by Smith

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480 routing games

(1979). Correa et al. (2004) proved Theorem 18.21 without any convexity assump-tions. This theorem has been generalized to wider classes of nonatomic games; seeRoughgarden (2005a) for a survey, as well as a discussion of the price of anarchy ofnonatomic selfish routing games with nonutilitarian objectives.

Finally, Theorem 18.29 is due to Roughgarden and Tardos (2002). A proof ofCorollary 18.30 and a counterexample to Theorem 18.29 in atomic instances can befound in Roughgarden (2005a). For extensions of Theorem 18.29 to networks withrestricted cost functions, including a solution to Exercise 18.8(e), see Chakrabarty(2004) and Correa et al. (2005).

18.6.2 Atomic Selfish Routing

Atomic selfish routing games were first considered by Rosenthal (1973), who provedTheorem 18.12, using the potential function method. Rosenthal also introduced the con-cept of “congestion games” (Remark 18.2). Monderer and Shapley (1996) undertook amore general study of “potential games” – games that admit a potential function, whichin turn can be used to prove that best-response dynamics converge to an equilibrium(Remark 18.13). Potential games are now studied in their own right; see Voorneveldet al. (1999) and Roughgarden (2005a, Section 4.8) for surveys of this literature.

Rosenthal (1973) showed that equilibrium flows need not exist in weighted multi-commodity atomic instances. Example 18.7 is due to Goemans et al. (2005). Fotakiset al. (2005) proved Theorem 18.15 for weighted instances with affine cost functions.

The price of anarchy of atomic instances was first studied by Suri et al. (2007) in thecontext of the asymmetric scheduling games described in Exercise 18.3 below. Amongother results, they proved an upper bound of 5/2 on the price of anarchy in such gameswhen each player controls one unit of traffic and when all cost functions are affine. Thispaper also introduced the proof structure used to prove Theorem 18.23 in this chapter.

Awerbuch et al. (2005) significantly generalized the results in Suri et al. (2007).They proved Theorem 18.23, as well as the refinement discussed in Exercise 18.7. TheAAE example and the variant in Exercise 18.2(a) are from Awerbuch et al. (2005),as are the exponential (in the degree bound p) upper and lower bounds on the priceof anarchy for polynomial cost functions with nonnegative coefficients. For refinedversions of these upper and lower bounds, see Olver (2006). Awerbuch et al. (2005)extended all of their upper bounds to mixed-strategy Nash equilibria. Goemans et al.(2005) extended the upper bounds to “sink equilibria,” a notion of equilibrium that ismotivated by best-response dynamics and that always exists in finite noncooperativegames.

For unweighted instances and pure-strategy equilibrium flows, the results inAwerbuch et al. (2005) were obtained independently by Christodoulou and Koutsoupias(2005b). The proofs in Christodoulou and Koutsoupias (2005b) extend without muchdifficulty to weighted instances and mixed-strategy Nash equilibria. Christodoulou andKoutsoupias (2005b) also studied the price of anarchy with respect to the egalitarianobjective (see Section 17.1) and provide solutions to parts (b) and (c) of Exercise 18.2.

Caragiannis et al. (2006) provide a solution to Exercise 18.3(b), as well as numerousother results about the price of anarchy and stability in different classes of asymmetricscheduling instances. For results on the price of stability in atomic selfish routing

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bibliography 481

games, see Anshelevich et al. (2004), Christodoulou and Koutsoupias (2005a), andCaragiannis et al. (2006).

Finally, several researchers have studied selfish routing in the atomic splittablemodel. This model is similar to the atomic selfish routing games studied in this chapter;the key difference is that a player i is permitted to route its ri units of traffic fractionallyover the si–ti paths of the network. This model is also different from nonatomic selfishrouting games; for example, if there is only one player controlling all of the traffic inthe network, then the player will minimize its cost by routing this traffic optimally.More generally, a player takes into account the congestion it causes for its own traffic,while ignoring the congestion it creates for other players.

Equilibrium flows in the atomic splittable model can behave in counterintuitive ways(see Exercise 18.9, taken from Catoni and Pallottino, 1991), and the price of anarchyin this model is not well understood. It was initially claimed that the upper boundson the price of anarchy for nonatomic instances carry over to atomic splittable ones(Correa et al., 2005; Roughgarden, 2005b), but Cominetti et al. (2006) recently gavecounterexamples to these claims in multicommodity networks. Obtaining tight boundson the price of anarchy in this model remains an important open question.

Bibliography

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E. Anshelevich, A. Dasgupta, J. Kleinberg, E. Tardos, T. Wexler, and T. Roughgarden. The price ofstability for network design with fair cost allocation. In Proceedings of the 45th Annual Symposiumon Foundations of Computer Science (FOCS), pp. 295–304, 2004.

D. P. Bertsekas. Nonlinear Programming, 2nd ed. Athena Scientific, 1999.M. J. Beckmann, C. B. McGuire, and C. B. Winsten. Studies in the Economics of Transportation.

Yale University Press, 1956.D. Braess. Uber ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung, 12:258–268,

1968. English translation in Braess (2005).D. Braess. On a paradox of traffic planning. Transportation Science, 39(4):446–450, 2005.D. Braess. http://homepage.ruhr-uni-bochum.de/Dietrich.Braess/, Homepage, 2007.D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation: Numerical Methods, 2nd

ed. Prentice-Hall, 1989. Athena Scientific, 1997.R. Cominetti, J. R. Correa, and N. E. Stier Moses. Network games with atomic players. In Proceedings

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J. R. Correa, A. S. Schulz, and N. E. Stier Moses. On the inefficiency of equilibria in congestion games.In Proceedings of the 11th Conference on Integer Programming and Combinatorial Optimization(IPCO), pp. 167–181, 2005.

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Exercises

18.1 Recall the nonlinear variant of Pigou’s example (Example 18.3). Prove that as thedegree p of the cost function of the second link tends to infinity, the price ofanarchy tends to infinity as p/ ln p.

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18.2 This exercise explores lower bounds on the price of anarchy in atomic selfishrouting games with affine cost functions.

(a) Modify the players’ weights in the AAE example (Example 18.6) so that the priceof anarchy in the resulting weighted atomic instance is precisely (3 + √

5)/2 ≈2.618.

(b) Can you devise an unweighted atomic instance with 3 players, affine costfunctions, and price of anarchy equal to 5/2? Can you achieve a price ofanarchy of (3 + √

5)/2 using 3 players and variable weights?(c) What is the largest price of anarchy in an atomic instance with affine cost

functions and only 2 players?

18.3 An asymmetric scheduling instance differs from an atomic selfish routing instance inthe following two respects. First, the underlying network is restricted to a commonsource vertex s, a common sink vertex t, and a set of parallel links that connect sto t. On the other hand, we allow different players to possess different strategy sets:each player i has a prescribed subset Si of the links that it is permitted to use.

(a) Show that every asymmetric scheduling instance is equivalent to an atomicselfish routing game. Your reduction should make use only of the cost functionsof the original scheduling instance, plus possibly the all-zero cost function.

(b) [Difficult] Part (a) shows that the worst-case price of anarchy in asymmetricscheduling instances with affine cost functions is at most that in atomic selfishrouting games with affine cost functions. Prove that the worst-case price ofanarchy is the same in the two models, equal to 5/2 in unweighted instancesand (3 + √

5)/2 in weighted instances.

18.4 Prove Theorem 18.15. Make use of the following potential function:

�( f ) =∑e∈E

⎛⎝ce( fe) fe +

∑i∈Se

ce(ri )ri

⎞⎠ ,

where Se denotes the set of players that choose a path in f that includes the edge e.

18.5 A set C of cost functions is inhomogeneous if it contains at least one functionc satisfying c(0) > 0. Extend Proposition 18.19 to inhomogeneous sets of costfunctions.

[Hint: Simulate a Pigou-like example using a more complex network and costfunctions drawn only from the given set C.]

18.6 Prove that if C is the set of nonnegative, nondecreasing, concave cost functions,then the Pigou bound α(C) equals 4/3.

18.7 Improve the upper bound of Theorem 18.23 for unweighted atomic instanceswith affine cost functions. Can you match the lower bound provided by the AAEexample?

18.8 This exercise studies refinements and extensions of Theorem 18.29.

(a) Deduce Corollary 18.30 from Theorem 18.29.(b) Show that Theorem 18.29 does not always hold in atomic selfish routing games.(c) Suppose we define f ∗ to be a flow feasible for the instance (G, (1 + δ)r, c),

where δ > 0 is a parameter. (In Theorem 18.29, δ = 1.) How does the guaranteeof Theorem 18.29 change?

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(d) Use Example 18.3 to prove that your bound in part (c) is the best possible.(e) Determine the smallest value of δ such that the following statement is true: for

every nonatomic instance (G, r, c) with affine cost functions, for every equilib-rium flow f for (G, r, c) and optimal flow f ∗ for (G, (1 + δ)r, c), C ( f ) ≤ C ( f ∗).(Theorem 18.29 implies that the statement holds with δ = 1; the question iswhether or not our restriction on the cost functions permits smaller values of δ.)

18.9 Recall the atomic splittable selfish routing model discussed at the end ofSection 18.6. Given such a game, we can obtain a new game by replacing aplayer that routes ri units of traffic from si to ti by two players that each route ri /2units of traffic from si to ti . This operation does not change the cost of an optimalflow. Intuitively, since it decreases the amount of cooperation in the network, itshould only increase the cost of an equilibrium flow. Prove that this intuition isincorrect: in multicommodity atomic splittable selfish routing networks, splitting aplayer in two can decrease the price of anarchy.

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